Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.

a. Describe situations in which opposite quantities combine to make 0. For example, a hydrogen atom has 0 charge because its two constituents are oppositely charged.

b. Understand p + q as the number located a distance |q| from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing real-world contexts.

c. Understand subtraction of rational numbers as adding the additive inverse, p – q = p + (–q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts.

d. Apply properties of operations as strategies to add and subtract rational numbers.

Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.

a. Describe situations in which opposite quantities combine to make 0. For example, a hydrogen atom has 0 charge because its two constituents are oppositely charged.

b. Understand p + q as the number located a distance |q| from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing real-world contexts.

c. Understand subtraction of rational numbers as adding the additive inverse, p – q = p + (–q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts.

d. Apply properties of operations as strategies to add and subtract rational numbers.

TEACHING CHANNEL
INTERVIEW WITH SABRINA JOSEPH
SABRINA JOSEPH:
We read about this the other day. Let’s recap it. OK. "Sometimes the chefs wanted to change the temperature of the soup without adding any cubes in. To do this they simply used their large ladle and scooped out hot or cold cubes. If they wanted to warm the soup up, they scooped out a..." A cold cube. If you want to warm something up, and there's some cold in it, we can scoop it out.
(interview)
My name is Sabrina Joseph. I teach seventh grade at Columbia Secondary School for Math, Science and Engineering in Harlem.
(class)
If they wanted to cool the soup down, they scooped out a...a hot cube.
(interview)
Today’s aim was, how do we subtract integers? They heard about the chef's soup story in a previous lesson, so they know the idea of these hot and cold cubes being used to change the temperature of the soup. Putting hot cubes in warms it up; putting cold cubes in cools the soup down. So then we looked at, well, if they don't want to add anything in, how can they warm the soup? And right in the beginning we said, well, I can take out cold to warm the soup up, or I can take out hot to cool the soup down.
(class)
What operation do you think taking out cubes is? Zach.
ZACH:
Oh, subtraction.
SABRINA JOSEPH:
Subtraction, OK. So then what do we call each of these? Like, a hot and a cold? What were they again? Karimeh?
KARIMEH:
A zero pair.
SABRINA JOSEPH:
A zero pair. So if we wanted to be able to take out two hot cubes and they weren't in there, we'd put in two zero pairs.
(class)
I noticed when I was working through the worksheet, there becomes that point when you have negative five and you need to take away negative nine. You don't have enough negative to take away. So I wanted to make sure I did an example with them that would let them see, oh, I can add in these zero pairs, one hot and one cold, that won't change what I’m starting with but will give me the extra negatives I need so that I can actually do the subtraction with the chips. So I did that example with them so that they'd be able to do that when it got to their worksheet.
(class)
What you're gonna do, I’m gonna give a cup of chips for each table, and then in pairs you're gonna model each cooking action that's on the worksheet. Then you're gonna write your number sentence for it, and then record the overall temperature change, your answer.
(interview)
We use manipulatives, we use the double-sided red and yellow chips. So they decided that the red side should represent the hot cube, or positive one, and then the yellow side of the chip would represent a cold cube, or negative one.
(class)
You guys have the chips. Use them!
(interview)
Working with a group of four, one person would have done everything, or two people would have done everything, and the rest would have just sat back. So I wanted them to work in pairs so that they could have somebody to fall back on, to help them, but not so that they couldn't do anything at all.
STUDENT 1:
For example, let's use this one. We have the yellows, minus four, and then we have, you have these three.
STUDENT 2:
That’s what I’m doing, I’m subtracting negative nine from negative five to get four.
SABRINA JOSEPH:
They use the chips to model, OK, I'm putting in five cold cubes, so that's negative five. I want to take out nine cold cubes. They don't have nine cold cubes, so they're putting in zero pairs, and they're actually working with the numbers in a hands-on kind of way to see what's really happening. Some of the students noticed quickly that, well, if we want to warm soup up, we don't have to just take cold out, we can add warm in. And so by the end, they get to this point where they're like, oh, taking out cold is the same as adding hot. So subtracting a negative is the same as adding in a positive, which is a concept that kids always get tripped up on. So I like this a lot for seeing that clearly.
(class)
So what's negative four minus three? But how are you getting seven? Seven looks bigger than four.
STUDENT 1:
No, but you're negative numbers.
SABRINA JOSEPH:
OK. All right, so you have a good rule. Why does it work?
(interview)
I gave them really like ten minutes to work on the worksheet with the manipulatives, and then come up with a rule as a group that they think could be used when they want to subtract all the time. Some of them had a rule already but didn't know why their rule worked. And then some of them really discovered the rule while they were working through this. After all of that, we come back together as a class to talk about, well, what is this rule?
STUDENT 3:
That’s what I put for my rule, which is that you add the opposite of the second number.
SABRINA JOSEPH:
I like the way Annie put her rule, so that's what we're gonna write. To subtract integers...
(interview)
No matter what we're subtracting, we can always add the opposite of that second number to have the same effect, and then we wrote the rule out that they'd be able to use. At this age, kids are really concrete, so they need to see things and feel things and touch things. So it's not a rule that I tell them, keep change change, which we hear all the time, for how do we subtract integers, but instead, they'll remember using the chips, they'll remember how we got to our answer, and if they ever forget what's the rule, they can always say, well, taking out cold cubes made the soup warmer, I can add in hot cubes and make the soup warmer. And they can always have that to go back to, so they're not stuck with, what's that rule that I forgot?
(class)
What operation can we say we're really doing then? Well what are we adding? Because we can add negative nine, that would have made it colder...
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